Units and Measurements • Topic 1 of 3

Units & the SI System

Physics is built on measurement. To measure a quantity is to compare it against a fixed, agreed reference called a unit, and report how many times that unit fits into the quantity. A physical quantity always has two parts: a numerical value and a unit. Writing the length of a room as just "5" is meaningless; $5\\,\\text{m}$ carries information. In symbols, a measured quantity $Q$ is written as $Q = n\\,u$, where $n$ is the numerical value and $u$ the unit. Crucially, $n$ and $u$ are inversely related: if you switch to a smaller unit, the number grows. The same length is $1\\,\\text{m}$ or $100\\,\\text{cm}$, so $n_1 u_1 = n_2 u_2$.

Quantities split into two groups. Fundamental (base) quantities are chosen to be independent of one another and cannot be expressed in terms of others — length, mass and time are the classic three. Derived quantities are built from the base ones by multiplication or division: speed is length divided by time, force is mass times acceleration, and so on. Their units are derived units assembled from base units.

A complete, self-consistent set of base units plus the derived units built from them is a system of units. Historically several competed — CGS (centimetre, gram, second), FPS (foot, pound, second) and MKS (metre, kilogram, second). The modern global standard is the SI system (Système International d'Unités), which rests on seven base units:

  • metre (m) for length, kilogram (kg) for mass, second (s) for time;
  • ampere (A) for electric current, kelvin (K) for thermodynamic temperature;
  • mole (mol) for amount of substance, candela (cd) for luminous intensity.

Two supplementary units handle angles: the radian (rad) for plane angle and the steradian (sr) for solid angle. To avoid clumsy numbers, SI uses prefixes that scale a unit by powers of ten — kilo ($10^{3}$), mega ($10^{6}$), giga ($10^{9}$), and downward milli ($10^{-3}$), micro ($10^{-6}$), nano ($10^{-9}$). So $1\\,\\text{km} = 10^{3}\\,\\text{m}$ and $1\\,\\mu\\text{m} = 10^{-6}\\,\\text{m}$.

SI also fixes conventions that keep scientific writing unambiguous: unit symbols are never pluralised (write $5\\,\\text{kg}$, not $5\\,\\text{kgs}$), no full stop follows a symbol unless it ends a sentence, a space separates the number from the unit, and symbols named after people are capitalised (N for newton, Pa for pascal) while the spelled-out name stays lower case.

Deeper Insight — why the kilogram was redefined: Until 2019 the kilogram was a lump of platinum-iridium kept in Paris, and the world's mass standard could literally lose atoms. The SI base units are now defined through fixed values of fundamental constants — the second via the caesium atom's transition frequency, the metre via the speed of light, and the kilogram via Planck's constant $h$. The lesson for you is conceptual: a good unit must be reproducible anywhere and unchanging in time, which is exactly why nature's constants make better rulers than physical objects.

The seven SI base unitsThe 7 SI Base UnitsLengthmetre (m)Masskilogram (kg)Timesecond (s)Currentampere (A)Temperaturekelvin (K)Substancemole (mol)Luminous Intensitycandela (cd)SI prefixes scale a unit by powers of tenCommon SI Prefixes (powers of ten)nano10^-9micro10^-6milli10^-3kilo10^3mega10^6giga10^9
1
Worked Example
Convert a speed of $72\\,\\text{km h}^{-1}$ into $\\text{m s}^{-1}$.
Solution
  1. $1\\,\\text{km} = 1000\\,\\text{m}$ and $1\\,\\text{h} = 3600\\,\\text{s}$.
  2. Multiply by the conversion factor: $72 \\times \\dfrac{1000}{3600}$.
  3. $\\dfrac{1000}{3600} = \\dfrac{5}{18}$, so $72 \\times \\dfrac{5}{18} = 20$.

Answer: $72\\,\\text{km h}^{-1} = 20\\,\\text{m s}^{-1}$.

2
Worked Example
Express $1\\,\\text{pascal}$ in terms of SI base units, given pressure $=$ force/area.
Solution
  1. Force in base units: $1\\,\\text{N} = 1\\,\\text{kg m s}^{-2}$.
  2. Area is in $\\text{m}^2$, so pressure $= \\dfrac{\\text{kg m s}^{-2}}{\\text{m}^2}$.
  3. Simplify: $\\text{kg m}^{-1}\\text{s}^{-2}$.

Answer: $1\\,\\text{Pa} = 1\\,\\text{kg m}^{-1}\\text{s}^{-2}$.

3
Worked Example
The radius of a hydrogen atom is about $0.53\\,\\text{angstrom}$. Express this in nanometres ($1\\,\\text{angstrom} = 10^{-10}\\,\\text{m}$).
Solution
  1. Convert to metres: $0.53 \\times 10^{-10}\\,\\text{m} = 5.3 \\times 10^{-11}\\,\\text{m}$.
  2. $1\\,\\text{nm} = 10^{-9}\\,\\text{m}$, so divide by $10^{-9}$.
  3. $\\dfrac{5.3 \\times 10^{-11}}{10^{-9}} = 5.3 \\times 10^{-2} = 0.053$.

Answer: $0.53\\,\\text{angstrom} = 0.053\\,\\text{nm}$.

4
Worked Example
A car covers $1\\,\\text{km}$ using $0.05\\,\\text{L}$ of fuel. Express the fuel consumption in $\\text{m}^3$ per metre of SI base units.
Solution
  1. $0.05\\,\\text{L} = 0.05 \\times 10^{-3}\\,\\text{m}^3 = 5 \\times 10^{-5}\\,\\text{m}^3$.
  2. $1\\,\\text{km} = 10^{3}\\,\\text{m}$.
  3. Consumption $= \\dfrac{5 \\times 10^{-5}}{10^{3}} = 5 \\times 10^{-8}\\,\\text{m}^3\\,\\text{m}^{-1}$.

Answer: $5 \\times 10^{-8}\\,\\text{m}^2$ (volume per unit length).

5
Worked Example
If the unit of length is doubled, by what factor does the numerical value of a fixed area change?
Solution
  1. Area has units of (length)$^2$, so $n_1 u_1 = n_2 u_2$ becomes $n_1 L^2 = n_2 (2L)^2$.
  2. $n_2 = n_1 \\dfrac{L^2}{4L^2} = \\dfrac{n_1}{4}$.

Answer: The numerical value becomes one-fourth, since a larger unit gives a smaller number.

6
Worked Example
The density of water is $1000\\,\\text{kg m}^{-3}$. Express it in $\\text{g cm}^{-3}$.
Solution
  1. $1\\,\\text{kg} = 10^{3}\\,\\text{g}$ and $1\\,\\text{m}^3 = (10^{2}\\,\\text{cm})^3 = 10^{6}\\,\\text{cm}^3$.
  2. $1000\\,\\text{kg m}^{-3} = 1000 \\times \\dfrac{10^{3}\\,\\text{g}}{10^{6}\\,\\text{cm}^3}$.
  3. $= 1000 \\times 10^{-3} = 1\\,\\text{g cm}^{-3}$.

Answer: $1000\\,\\text{kg m}^{-3} = 1\\,\\text{g cm}^{-3}$.

Key Points

  • Every physical quantity is a numerical value times a unit; $n$ and $u$ are inversely related, so $n_1 u_1 = n_2 u_2$.
  • Fundamental quantities (length, mass, time, etc.) are independent; derived quantities are built from them.
  • The SI system has seven base units: m, kg, s, A, K, mol, cd, plus radian and steradian as supplementary units.
  • Prefixes scale units by powers of ten: kilo $10^{3}$, mega $10^{6}$, milli $10^{-3}$, micro $10^{-6}$, nano $10^{-9}$.
  • SI conventions: symbols are not pluralised, a space separates value and unit, person-named symbols are capitalised (N, Pa, K).
Tap an option to check your answer0 / 4
Q1.Which of these is NOT an SI base unit?
Explanation: The newton is a derived unit ($\\text{kg m s}^{-2}$); kelvin, mole and candela are base units.
Q2.$72\\,\\text{km h}^{-1}$ equals how many $\\text{m s}^{-1}$?
Explanation: Multiply by $\\tfrac{5}{18}$: $72 \\times \\tfrac{5}{18} = 20\\,\\text{m s}^{-1}$.
Q3.The prefix 'micro' stands for a factor of:
Explanation: Micro means $10^{-6}$; milli is $10^{-3}$ and nano is $10^{-9}$.
Q4.If the unit of a quantity is made larger, the numerical value of a fixed measurement:
Explanation: Since $n \\propto 1/u$, a larger unit gives a smaller numerical value.